LAGRANGE’S EQUATION OF MOTION

In the case of dynamic analysis of structures, the direct application of the well-known Lagrange’s equation of motion can be used to develop the dynamic equilibrium of a complex structural system. Lagrange’s minimization equation, written in terms of the previously defined notation, is given by:

                                                        (1)

The node point velocity is defined as  . The most general form for the kinetic energy  stored within a three-dimensional element i of mass density  is:

               (2)

The same shape functions used to calculate the strain energy within the element allow the velocities within the element to be expressed in terms of the node point velocities. Or:

  ,   , 

Therefore, the velocity transformation equations can be written in the following form:

Using exact or numerical integration, it is now possible to write the total kinetic energy within a structure as:

The total mass matrix  is the sum of the element mass matrices . The element consistent mass matrices are calculated from:

                                                   (3)

where  is the 3 by 3 diagonal mass density matrix shown in Equation (2).

Equation (3) is very general and can be used to develop the consistent mass matrix for any displacement-based finite element. The term “consistent” is used because the same shape functions are used to develop both the stiffness and mass matrices.

Direct application of Equation (1) will yield the dynamic equilibrium equations: